The Propagating Power of Similarity. The relation of similarity in all its degrees is reciprocal. So far as things are alike, either may be substituted for the other; and this may perhaps be considered the very meaning of the relation. But it is well worth notice that there is in similarity a peculiar power of extending itself among all the things which are similar. To render a number of things similar to each other we need only render them similar to one standard object. Each coin struck from a pair of dies not only resembles the matrix or original pattern from which the dies were struck, but resembles every other coin manufactured from the same original pattern. Among a million such coins there are not less than 499,999,500,000 pairs of coins resembling each other. Similars to the same are similars to all. It is one great advantage of printing that all copies of a document struck from the same type are necessarily identical each with each, and whatever is true of one copy will be true of every copy. Similarly, if fifty rows of pipes in an organ be tuned in perfect unison with one row, usually the Principal, they must be in unison with each other. Similarity can also reproduce or propagate itself ad infinitum: for if a number of tuning-forks be adjusted in perfect unison with one standard fork, all instruments tuned to any one fork will agree with any instrument tuned to any other fork. Standard measures of length, capacity, weight, or any other measurable quality, are propagated in the same manner. So far as copies of the original standard, or copies of copies, or copies again of those copies, are accurately executed, they must all agree each with every other. It is the capability of mutual substitution which gives such great value to the modern methods of mechanical construction, according to which all the parts of a machine are exact facsimiles of a fixed pattern. The rifles used in the British army are constructed on the American interchangeable system, so that any part of any rifle can be substituted for the same part of another. A bullet fitting one rifle will fit all others of the same bore. Sir J. Whitworth has extended the same system to the screws and screw-bolts used in connecting together the parts of machines, by establishing a series of standard screws. Anticipations of the Principle of Substitution. In such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained. The germ at least of every doctrine will be found in earlier writers, and novelty must arise chiefly in the mode of harmonising and developing ideas. When I first employed the process and name of substitution in logic, I was led to do so from analogy with the familiar mathematical process of substituting for a symbol its value as given in an equation. In writing my first logical essay I had a most imperfect conception of the importance and generality of the process, and I described, as if they were of equal importance, a number of other laws which now seem to be but particular cases of the one general rule of substitution. . My second essay, "The Substitution of Similars," was written shortly after I had become aware of the great simplification which may be effected by a proper application of the principle of substitution. I was not then acquainted with the fact that the German logician Beneke had employed the principle of substitution, and had used the word itself in forming a theory of the syllogism. My imperfect acquaintance with the German language had prevented me from acquiring a complete knowledge of Beneke's views; but there is no doubt that Professor Lindsay is right in saying that he, and probably other logicians, were in some degree familiar with the principle. Even Aristotle's dictum may be regarded as an imperfect statement of the principle of substitution; and, as I have pointed out, we have only to modify that dictum in accordance with the quantification of the predicate in order to arrive at the complete 1 Pure Logic, pp. 18, 19. Ueberweg's System of Logic, transl. by Lindsay, pp. 442-446, 571, 572. The anticipations of the principle of substitution to be found in the works of Leibnitz, Reusch, and perhaps other German logicians, will be noticed in the preface to this second edition. process of substitution.1 The Port-Royal logicians appear to have entertained nearly equivalent views, for they considered that all moods of the syllogism might be reduced under one general principle. Of two premises they regard one as the containing proposition (propositio continens), and the other as the applicative proposition. The latter proposition must always be affirmative, and represents that by which a substitution is made; the former may or may not be negative, and is that in which a substitution is effected. They also show that this method will embrace certain cases of complex reasoning which had no place in the Aristotelian syllogism. Their views probably constitute the greatest improvement. in logical doctrine made up to that time since the days of Aristotle. But a true reform in logic must consist, not in explaining the syllogism in one way or another, but in doing away with all the narrow restrictions of the Aristotelian system, and in showing that there exists an infinite variety of logical arguments immediately deducible from the principle of substitution of which the ancient syllogism forms but a small and not even the most important part. The Logic of Relatives. There is a difficult and important branch of logic. which may be called the Logic of Relatives. If I argue, for instance, that because Daniel Bernoulli was the son of John, and John the brother of James, therefore Daniel was the nephew of James, it is not possible to prove this conclusion by any simple logical process. We require at any rate to assume that the son of a brother is a nephew. A simple logical relation is that which exists between properties and circumstances of the same object or class. But objects and classes of objects may also be related according to all the properties of time and space. I believe it may be shown, indeed, that where an inference concerning such relations is drawn, a process of substitution is really employed and an identity must exist; 1 Substitution of Similars (1869), p. 9. 2 Port-Royal Logic, transl. by Spencer Baynes, pp. 212-219. Part III. chap. x. and xi. but I will not undertake to prove the assertion in this work. The relations of time and space are logicalrelations of a complicated character demanding much abstract and difficult investigation. The subject has been treated with such great ability by Peirce,' De Morgan,2 Ellis, and Harley, that I will not in the present work attempt any review of their writings, but merely refer the reader to the publications in which they are to be found. 1 Description of a Notation for the Logic of Relatives, resulting from an Amplification of the Conceptions of Boole's Calculus of Logic. By C. S. Peirce. Memoirs of the American Academy, vol. ix. Cambridge, U.S., 1870. 2 On the Syllogism No IV., and on the Logic of Relations. By Augustus De Morgan. Transactions of the Cambridge Philosophical Society, vol. x. part ii., 1860. 3 Observations on Boole's Lows of Thought. By the late R. Leslie Ellis; communicated by the Rev. Robert Harley, F.R.S. Report of the British Association, 1870. Report of Sections, p. 12. Also, On Boole's Laws of Thought. By the Rev. Robert Harley, F.R.S., ibid. p. 14. CHAPTER II. TERMS. EVERY proposition expresses the resemblance or difference of the things denoted by its terms. As inference treats of the relation between two or more propositions, so a proposition expresses a relation between two or more terms. In the portion of this work which treats of deduction it will be convenient to follow the usual order of exposition. We will consider in succession the various kinds of terms, propositions, and arguments, and we commence in this chapter with terms. The simplest and most palpable meaning which can belong to a term consists of some single material object, such as Westminster Abbey, Stonehenge, the Sun, Sirius, &c. It is probable that in early stages of intellect only concrete and palpable things are the objects of thought. The youngest child knows the difference between a hot and a cold body. The dog can recognise his master among a hundred other persons, and animals of much lower intelligence know and discriminate their haunts. In all such acts there is judgment concerning the likeness of physical objects, but there is little or no power of analysing each object and regarding it as a group of qualities. The dignity of intellect begins with the power of separating points of agreement from those of difference. Comparison of two objects may lead us to perceive that they are at once like and unlike. Two fragments of rock may differ entirely in outward form, yet they may have the same colour, hardness, and texture. Flowers which agree in colour may differ in odour. The mind learns to regard |